3 Answers
3

There is a recent construction of a fully extended 4d TQFT from a modular tensor category,
due to Dan Freed and Constantin Teleman (using Lurie's proof of the cobordism hypothesis).
It is described in Freed's lecture notes from the Segal 70th birthday conference here:
https://people.maths.ox.ac.uk/tillmann/ASPECTS.html

The idea is that braided tensor categories are naturally objects of a "Morita" 4-category
(morphisms are algebra objects in bimodule categories, 2-morphisms
are bimodules categories for these, 3-morphisms are functors of those,
and 4-morphisms are natural transformations --- the quick mnemonic is
that braided counts for two, category counts for one, together we get three,
and three-categories form a four-category ---- a baby version of this is
that algebras form a two-category, while monoidal categories (algebras in categories)
form a three category).

Freed and Teleman show that modular categories are "superduper finite" (aka fully dualizable)
objects of this category, ie satisfy the conditions of the cobordism hypothesis
to define a functor from the 4d-bordism category. In fact much more is true -- this
field theory is an invertible field theory... basically it means it's completely characterized by a single characteristic class of four manifolds, the "anomaly" of the original modular tensor category.

So in fact you shouldn't think of this 4d field theory as more information --it's LESS information than the 3d field theory attached to the MTC, but rather it's the anomaly information
needed to completely define the three-dimensional field theory (which they
use to extend Chern-Simons theory to a point eg.)

Edit: As a result of some interesting exchanges with Kevin Walker and Dan Freed I
believe things are a little more complicated than I had initially understood.
The results of Freed-Teleman indeed imply that the 4d CYK TFT is an invertible field theory,
i.e. that modular tensor categories are invertible objects in the Morita 4-category
of braided tensor categories. This means that the entire field theory can be described by a map
of spectra -- namely the sphere spectrum (classifying space of the framed cobordism category) mapping to the space of invertible objects in the Morita category. However, it's not clear exactly what this target space IS --- what's much easier to see I believe is the OTHER space attached
to the Morita category, namely its classifying space (where we invert morphisms to make a groupoid, rather than restrict to invertible morphisms as well as invertible objects). The latter map
is close to the classical notion of anomaly as far as I understand, but the map
that truly classifies modular tensor categories up to Morita equivalence is the former,
about which it appears not much is known.

When you say it's less info is that because you're talking about the 34 theory not the 234 theory? The oriented 234 theory is exactly the same info as the anomalous 23 theory, right?
–
Noah SnyderJan 18 '13 at 14:55

3

@Noah: no, I'm talking about the 01234 theory and it's still less info. the anomalous 23 (or 123) Reshetikhin-Turaevish theory is VALUED in the invertible field theory that Dan and Constatin construct, but that latter theory is completely determined by a single characteristic class, and just keeps track of the actual anomaly, not of the anomalous theory.
–
David Ben-ZviJan 18 '13 at 16:52

The TQFT in question should probably be called the Crane-Yetter or Crane-Yetter-Kauffman TQFT.
Crane-Yetter-Kauffman didn't work it out as a fully extended theory, and didn't notice (so far as I can tell) the relation to Witten-Reshetikhin-Turaev theories, but they definitely were the first to write down the 4d part of the theory.

The CYK TQFT contains all of the information of the WRT TQFT. (This disagrees with David Ben Zvi's answer, but I think the difference is due to our using different axiomatic frameworks for TQFTs, not disagreement about mathematical facts.) More specifically,
$$
Z_{WRT}(X, \Gamma) = Z_{CYK}(\partial^{-1}(X); \Gamma).
$$
Here $X$ is a manifold of dimension 1, 2 or 3 (not necessarily closed). $X$ is equipped with extra structure ($p_1$ structure, signature structure, null-bordism structure, ...) which makes $\partial^{-1}(X)$ sufficiently unambiguous. (For example, if $X$ is a closed 3-manifold, then the choice of $\partial^{-1}(X)$ only matters up to bordism.) The $\Gamma$ on the left hand side is a collection of "Wilson loops" or more generally a Wilson (labeled) graph. The $\Gamma$ on the right hand side is a boundary condition. (Same graph, but different interpretation.)

One way of looking at this is as follows. We expect, roughly, a correspondence
$$
\mbox{$n$-category} \;\; \leftrightarrow \;\; \mbox{$(n{+}1)$-dimensional TQFT}.
$$
The input data for for a WRT TQFT is a modular tensor category, which is a particular type of 3-category. But the WRT TQFT is a (2+1)-dimensional theory, not a (3+1)-dimensional theory, so something weird is going on here. The natural thing to do with a modular tensor category is to build the (3+1)-dimensional CYK TQFT, which is fully extended (a "0-1-2-3-4" theory) and anomaly-free. One then notices that the CYK theory is almost trivial for closed manifolds (more specifically, the dimensional reduction by $S^1$ is 2-Morita trivial), so we can derive from the CYK TQFT the (2+1)-dimensional WRT TQFT via the slogan
$$
Z_{WRT}(X, \Gamma) = Z_{CYK}(\partial^{-1}(X); \Gamma).
$$
But note that since CYK is merely almost trivial and not completely trivial, the WRT TQFT acquires an anomaly (i.e. manifolds need to be equipped with extra structure). Also, since it's hard to make sense of $\partial^{-1}(X)$ when $X$ is a point, the WRT theory is not fully extended; it's a 1-2-3 theory rather than a 0-1-2-3 theory.

I should also note that the input data for the CYK can be a premodular category ($S$-matrix perhaps degenerate). When the input is premodular but not modular, then the CYK TQFT is not almost trivial and we cannot construct a (2+1)-dimensional TQFT as above.

@Kevin: very interesting! definitely had missed that point. I'm confused though since the 4d TFT only depends on your modular tensor category C up to Morita equivalence - are you saying it's Morita rigid? Or put another way, I think you're saying we recover C=Z_{CYK}(disc) - but I only see the latter as an object in D=Z_{CYK}(S^1), and unless I give the extra structure of forgetful functor from D to categories how do I recover the underlying category of C? That forgetful functor (domain wall) seems to me equivalent to the data of the anomalous theory Z_{WRT} itself, and not part of Z_{CYK}?
–
David Ben-ZviMay 16 '13 at 15:28

@David: I'm not sure I fully understand your question but I'll answer as best I can. I would say that $C$, or rather $Rep(C)$, is isomorphic to $Z_{CYK}(pt)$, not $Z_{CYK}(D^2)$. Perhaps part of the confusion is due to the fact that my TQFT framework is not the Atiyah-Segal framework. Compared to Atiyah-Segal, I have some extra structure at my disposal: "fields" or boundary conditions on manifolds. (The graph $\Gamma$ above is an example of such.) My Hilbert spaces are realized concretely as functions on boundary conditions. There is often...
–
Kevin WalkerMay 16 '13 at 15:58

[continued] ... a preferred boundary condition corresponding to gluing together many copies of (iterated) identity morphisms of the input $n$-category, and evaluation at this preferred boundary condition gives a preferred map from Hilbert spaces to the ground ring. If W is a 4-manifold with boundary, then applying this preferred map to the element $Z_{CYK}(W)$ of the Hilbert space gives an element of the ground ring which is equal to $Z_{WRT}(\partial(W))$.
–
Kevin WalkerMay 16 '13 at 16:04

1

I think we're simply working with different amounts of structure. In the Freed-Teleman setting (via cobordism hypothesis) Z_{CYK}(pt) is indeed C, but as an object of the Morita category, so is equivalent to any other modular tensor category with the same anomaly invariant. So in that setting you can't functorially recover C (hence WRT) itself from the CYK TFT, but only a single characteristic class. On the other hand if you give enough additional structure, you can rigidify from the Morita category to the category of braided tensor categories and then recover C.
–
David Ben-ZviMay 16 '13 at 16:16

For a parallel example: given a (2-dualizable) associative algebra A, considered as an object of the Morita 2-category, we can construct a 2d TFT Z. But you can only recover the category A-mod functorially from Z, or A up to Morita equivalence, even though one could say A is Z(pt).. to see A itself you need extra structure: a specific boundary condition in the field theory Z, or equivalently a (compact) object of A-mod. In the Freed-Teleman language, that defines a theory relative to the 2d TFT Z, analogous to WRT being relative to CYK, and that relative theory is what knows A itself.
–
David Ben-ZviMay 16 '13 at 16:20

Originally, the idea of a 4D TQFT was to be found in a Hopf Category as defined by Crane and Igor Frenkel. Crane and Yetter gave an example via certain cocycles over a finite group. Kauffman, Saito, and I explicitly constructed this, but never were able to compute with it.

One should also look through Lurie's work to get explicit examples of braided monoidal 2-categories with duals. From David Ben-Zvi's description above, I would guess these are Morita 4 categories.